As described above, the full noise model we use involves two noise parameters, the noise variance \(\delta \)  and the swap-out probability \(p_0\) ; the error threshold is a line in \((\delta , p_0)\)  parameter space rather than a single point.
To estimate this error threshold, we run Monte Carlo simulations as described in alg:ft for different values of \(\delta \) , \(p_0\)  and \(d\)  (the lattice size).
For a particular value of \(p_0\) , we can extract the corresponding threshold \(\delta \)  value by plotting the logical error probabilities, \(p_{\rm fail}\) , for a range of values of \(\delta \)  and \(d\) .
The error threshold is then the point where curves for different \(d\)  intersect.
Equivalently, we can instead fix a value of \(\delta \)  and vary \(p_0\)  and \(d\) .
In the inner decoder we use standard binning, and we use matching graph weights derived from eq:errorprob in the outer decoder.
fig:thresholds shows the below-threshold region in \((\delta , p_0)\)  parameter space, alongside an example threshold plot for \(p_0=0.1\) .
We find a high tolerance to swap-outs, with a maximum swap-out threshold of \(p_0 \approx 0.236\)  (for \(\delta = 0\) ).
For \(p_0 = 0\) , the noise variance error threshold is \(\approx {10.5}{}\) , where the dB value is given by \(-10\log _{10}\delta \) .
As expected, an increase in the swap-out probability leads to an increase in the squeezing thresholds.
For an experimentally accessible [1]} squeezing value of \({15}{}\)  , our simulations suggest a swap-out threshold of \(p_0 \approx 0.133\) .
We note that the noise variance (\(\delta \) ) tolerance of our decoder is markedly better for \(p_0 \lesssim 0.19\)  than for values nearer the swap-out threshold.
Understanding this behaviour is an open problem, with one possible reason being that the inner and outer decoders we are using for the current simulations might be sub-optimal for this regime.
Therefore, to investigate this phenomenon further, we should compare our decoding strategy with e.g. maximum-likelihood decoding, in order to ascertain whether the sharp decrease in performance is a fundamental property or an artifact of our decoding strategy.
We leave this analysis for future work.
